The Minoans used columns in their palaces and the Mycenaeans used columns in their megarons which were inspired by the Minoan column, but were the columns of classical Greek temples and buildings a continuation of the Mycenaean columns or were they distinct from this tradition?

No. "There is a clear division between the architecture of the preceding Mycenaean culture and Minoan cultures and that of the ancient Greeks, the techniques and an understanding of their style being lost when these civilisations fell" From the Wikipedia page on Ancient Greek architecture
– sempaiscuba♦May 6 '19 at 19:49

1 Answer
1

No and yes. The 'continuity' was largely broken off with the collapse of Mycenean culture and it took too long for the Hellenes to re-reach the cultural level in architecture displayed in the bronze age Helladic cultures. They are hundreds of years apart. More than enough time for knowledge to get lost and exemplary buildings to be destroyed so far beyond recognition that reverse engineering became more difficult than starting anew. For quite some time, the Hellenes did not build in stone and is assumed that they were for a time just incapable to do that on a technological level.

What there is are geographical and structural similarities, like building big houses (megaron), in stone with hefty weight of the material, requiring columns for support. As these were sometimes built on old megarons, one might argue for continuity, but as they are structured differently above the foundations this is at least a bit questionable.

Classical Greek architectures are likely and mostly independent re-inventions, rough inspirations from rubbly remains. Most well known is here the fact that the Greeks just did not know how their 'heroes' (Myceneans) built with these gigantic stone blocks.

At a conventional date of 1350 BC, the fortifications on the acropolis, and other surrounding hills, were rebuilt in a style known as cyclopean because the blocks of stone used were so massive that they were thought in later ages to be the work of the one-eyed giants known as the Cyclopes (singular: Cyclops). Within these walls, much of which can still be seen, successive monumental palaces were built. The final palace, remains of which are currently visible on the acropolis of Mycenae, dates to the start of LHIIIA:2. Earlier palaces must have existed, but they had been cleared away or built over.–– Wikipedia: Mycenae Late Helladic III (LHIII; c.1400 – c.1050 BC)

That indicates an even more pronounced break if looking at that the very different techniques for later Greek buildings. Coupled with the lack of finds representing hundreds of intervening years, with much simpler economy, administration, religion and buildings throughout the Greek world.

But the argument for a continuity of certain parts is expressed as well:

The Hellenic sanctuary and its humbler counterpart, the rich man's house, are patently derived from the Mycenean palace compound, for which a precedent existed as Troy in the early Bronze age; nothing comparable is known during the intervening thousand years, but the resemblance seems 100 precise for mere coincidence. More surprising, because the hiatus may amount to two thousand years, is the parallel between the tvpe of hall then in vogue at Troy, with a projection behind like a smaller edition of the porch in front, and the Hellenic
temple with its opisthodomos balancing the porch. Except, however, for this one feature, we can watch the development of halls on the mainland of Greece from the same remote epoch, through successive innovations due to the invaders of the Middle Bronze Age and to Minoan influence upon the Myceneans, till the emergence of the temple.

Basically, then, Hellenic architecture is a synthesis of the pre-Hellenic styles. Even one of the past cultures had made its contribution, thanks probably to a diversity of racial and social factors as well as to differences in the physical environment. (Even the Early Bronze Age inhabitants of the Cyclades had bequeathed something, by showing how to construct a corbelled vault with the rock fragments that littered their barren islands.) The accumulated stock of architectural heritage reached its maximum at the close of the Bronze Age and became impoverished in the ensuing dismal centuries of transition, but the several types of building persisted; the loss was not in essentials but in technical and aesthetic standards, in decency of workmanship, and in elegance and ornament. When ease and wealth returned, and the Greeks learnt afresh how to build in stone, their first preoccupation was naturally to master the new technique, which they applied to reproducing the forms they were already accustomed to use in sun-dried brick and timber.–– A. W. Lawrence: "Greek Architecture" Revised by R. A. Tomlinson, Pelican History of Art, Yale University Press: New Haven, London, 51996.

This is break and continuity does not apply to all parts of the architectural language, style and technologies equally, though

Sources for the peristyle are equally difficult to pinpoint. Egyptian architecture is well known for the use of freestanding supports, either rectangular pillars or rounded columns. Yet these are often found in the interior of a structure, such as a hypostyle hall. The enclosure of a building by columns is unusual and the identification of a peristyle in extant examples, problematic (Haeny 2001). A derivation from Egypt is also undermined by chronology, since, as already discussed, the earliest known Greek peripteroi, the hekatompedon at Ano Mazaraki (circa 700 BCE) and the temple at Ephesos (second quarter of the seventh century BCE) precede the traditional time of contact.

According to A. Bammer, the Artemision peripteros had a predecessor, the “pre‐peripteros,” and its form raises further questions of foreign influence. It had columns in the same locations on the exterior and interior, but lacked walls. Bammer proposes a roof between the columns, leaving a central opening. Such an arrangement would recall the Mycenaean megaron with its central clerestory. On the other hand, with a continuous roof throughout, the pre‐peripteros would bear similarities to the Egyptian pavilion, which consisted of two rectangular but concentric series of pillar supports, also without walls. This type of structure is represented by the Kiosk of Sesostris I at Karnak (see Haeny 2001: 95–96 and fig. 12).

It was typical for Greek buildings of the Early Iron Age to be roofed in thatch and mud‐plaster. Although these perishable materials seldom survive, pitched or flat roofs are depicted in some drawings and models from the period (Schattner 1990). Terracotta tiled roofs appear in the archaeological record by the middle of the seventh century BCE. Although small, shingle‐like terracotta tiles were used much earlier to roof Early Helladic complexes such as the House of Tiles at Lerna, the tradition vanished by the Middle Helladic (Winter 1993: 8–9; Wiencke 2000: 197–307). Despite the rarity of examples, Mycenaeans seem to have developed interlocking tile roofs (Iakovides 1990). There is, however, no evidence for terracotta roof tiles after the Mycenaean collapse, and it is hard to imagine how a tile‐making tradition could have been kept alive in the relatively crude architecture of the intervening centuries (Wikander 1988: 204–205; Winter 1993: 13; Mazarakis Ainian 1997: 272). Thus, the surviving remains indicate that during the seventh century BCE, Greeks invented new systems of terracotta tiling to roof their most important sanctuary buildings.

In looking back over thousands of years of architectural history, however, it is clear that the external forms of classical architecture were dependent on both materials and methods of construction. While the Greek world had developed a monumental stone architectural tradition during the Bronze Age, as seen at Mycenaean citadels and Minoan palaces and villas, the demise of these cultures in the late second millennium left no practical legacy for subsequent generations. Visible remains, such as the Lion Gate at Mycenae, might have provided inspiration, but they did not offer practical advice. Most examples of temples or shrines dating between the twelfth and seventh centuries BCE were built of rubble masonry with earth mortar using techniques that are common to vernacular domestic architecture. For the ancient Greeks of the later, historical period, the stimulus to build in stone may have resulted from their own experience of seeing stone architecture in Egypt during the seventh and sixth centuries BCE. Monumental, often megalithic, architecture had been used for sacred buildings, including mortuary complexes and temples, for thousands of years in Egypt. Techniques used to quarry and dress stone and to assemble masonry may thus have been inspired by Egypt, but it seems likely that other factors contributed to the development of a distinctively Greek style.

The use of architectural terracottas in Greece also began in the second half of the seventh century and offered a durable, decorative material with which to roof a building and protect it from intemperate weather (see Chapter 4). The weight of such a roof, however, was significantly greater than anything used previously and was an additional stimulus to construction in stone (Rhodes 2003: 86–93). By the late seventh century BCE, the first intentionally designed stone architecture appeared in Greece. Extant remains from the early temples at Corinth and Isthmia display a variety of blocks that were cut to serve specific purposes, although they show little, if any, sign of an emergent architectural order (Rhodes 2003). In the generations that follow, these first efforts to build in stone led to the development of classical architecture.

In the tradition of so many Dark Age religious foundations that were inspired by the discovery of Mycenaean remains, the religious center of Athens was originally founded on the ruins of a Mycenaean palace high on the Acropolis rock. As late as the Periclean version of the Acropolis, physical contact with and display of the Mycenaean remains was critical to the nature of the sanctuary and its cults. The difference between Athens and early religious foundations in most of the rest of mainland Greece was that the Athenians were building on the architectural remains of Mycenaean ancestors who apparently had not been overcome and displaced by invasion, plague, or drought but who seem to have remained on the same spot continuously from the age of the heroes to the dawn of classical Greece. The Athenians were reminded of this every time they heard the Dorian dialects of their neighboring mainland city‐states, for the Athenians still spoke the ancient Ionian dialect of the Mycenaeans. The Athenians felt so deeply rooted into their native soil that they believed their first king, Kekrops, had slithered out of the earth, snaky tail and all, to found the city.

On the Periclean Acropolis, the great antiquity of the Athenians was preserved and displayed most explicitly in the cults and architecture of the Erechtheion, the successor to the Temple of Athena Polias, the most venerable of the temples on the Acropolis and originally built on the spot of the megaron of the Mycenaean palace. Housed in and immediately around the Erechtheion, in addition to the very ancient cult image of Athena, were many cults and artifacts associated with the mythical foundation of the city, including a cult to the snaky‐bodied king, the saltwater spring and olive tree of the contest between Athena and Poseidon for control of the original city, and the marks left by Poseidon’s trident in the bedrock when he struck down Erechtheus, another legendary Athenian king. The roots of the people of Athens were so deep and venerable that their ancient history was populated with heroes and gods (Figure 11.5).

The great Periclean gateway to the Acropolis, the Propylaia, and the Sanctuary of Athena Nike also enshrined and displayed the most ancient history of the Acropolis. The design of Mnesikles, the architect of the Propylaia, not only respected the best preserved and tallest stretch of Mycenaean wall on the Acropolis but it also actually incorporated it – for visual, not structural purposes – into its southeast corner. The new Propylaia acknowledged and celebrated its physical and spiritual connection with the earliest remains and history of the Acropolis through the intentional display of the city’s original fortification wall. Less obvious, as it is sheathed in fifth‐century ashlar masonry, is the incorporation of another Mycenaean structure, perhaps a defensive bastion, into the Periclean gateway complex. Originally freestanding in front of the Mycenaean wall but now connected with the Propylaia proper by classical masonry, the Mycenaean structure served as the structural base of a pedestal for a monument to victory, the small Temple of Athena Nike. A small window was left in the classical masonry near its base that allowed direct visual and tactile access to the cyclopean stones that formed the original Mycenaean foundation and, now, the classical one. Finally, less overt but equally expressive of Mycenaean palace architecture, the remains of the Mycenaean entranceway seem to have inspired Mnesikles to design the Propylaia according to Mycenaean principles of defensive architecture, with outstretched wings that encompassed and channeled anyone approaching, very much like the Mycenaean gateway to the Acropolis and the Lion Gate at Mycenae.

The Parthenon, too, might carry Mycenaean allusion. As the original temple of Athena Polias was founded on the ruins of the Mycenaean palace, and consistent with the universal fascination of early mainland Greeks with their heroic past, the name “Parthenon” may in fact indicate “the place of the maidens” rather than an epithet for Athena, and the western cella – the only part of the temple actually referred to by the ancients as the “Parthenon” – may reflect and preserve the position of the tomb of the daughters of one of the legendary founding fathers of Athens (Connelly 1996: 76). It was through the agency of their self‐sacrifice that Athens was first miraculously delivered from destruction at the hands of an invading enemy, deep in the shadows of mythical history.–– Margaret M. Miles (Ed): "A Companion to Greek Architecture", John Wiley & Sons: Chichester, 2016.

To highlight the specifically the column in a summary:

Why and how either style came to emerge in Greece, and why they came together into succinct systems so quickly, are questions that still puzzle scholars. Remains of the oldest surviving temples show that the main features of the Doric style were already established soon after 600 BCE. It is possible that the temple's central unit, the cella and porch, derived from the plan of the Mycenaean megaron, either through continuous tradition or by way of revival. If this is true, this relationship may reflect the revered place of Mycenaean culture in later Greek mythology.

Still, a Doric column shaft tapers upward, not downward like the Minoan-Mycenaean column. This recalls fluted half-columns in the funerary precinct of Djoser at Saqqara of over 2,000 years earlier. In fact, the very notion that temples should be built of stone and have large numbers of columns was an Egyptian one, even if Egyptian architects designed temples for greater internal traffic. Scholars believe that the rise of monumental stone architecture and sculpture must have been based on careful, on- the-spot study of Egyptian works and the techniques used to produce them. The opportunity for just such a study was available to Greek merchants living in trading camps in the western Nile Delta, by permission of the Egyptian king Psammetichus I (r. 664 610 BCE).

Some scholars see Doric architecture as a petrification (or turning to stone) of existing wooden forms, so that stone form follows wooden function. Accordingly, triglyphs once masked the ends of wooden beams, and the droplike shapes below, called guttae, mimick the wooden pegs that held them in place. Metopes evolved from boards that filled gaps between triglyphs to guard against moisture. Some derivations are more convincing than others, however. The vertical subdivisions of triglyphs hardly seem to reflect the forms of three half-round logs, as some have suggested, and column flutings need not have developed out of tool marks on a tree trunk, since Egyptian builders also fluted their columns and yet rarely used timber for supporting members. The question of how well function explains stylistic features faces the architectural historian repeatedly. –– Penelope E Davies et al (Eds): "Janson's History of Art: The Western Tradition", Prentice Hall: Upper Saddle River, London, 82011.